THE COPERNICUS OF GEOMETRY
two geodesies always intersect in precisely two points. Again,
on a plane, no two geodesies can enclose a space - as Euclid
assumed in one of the postulates for his geometry; on a sphere,
any two geodesies always enclose a space.
N
Imagine now the equator on the sphere and two geodesies
drawn through the north pole perpendicular to the equator. In
the northern hemisphere this gives a triangle with curved sides,
two of which are equal. Each side of this triangle is an arc of a
geodesic. Draw any other geodesic cutting the two equal sides
so that the intercepted parts between the equator and the cut-
ting line are equal. We now have, on the sphere, the four-sided
figure corresponding to the AXYB we had a few moments ago
in the plane. The two angles at the base of this figure are right
angles and the corresponding sides are equal, as before, but each
of the equal angles atX9Yi$ now greater than a Tight angle. So,
in the highly practical geometry of great circle sailing, which is
closer to real human experience than the idealized diagrams of
elementary geometry ever get, it is not Euclid's postulate
which is true - or its equivalent in the hypothesis of the right
angle - but the geometry which follows from the hypothesis of
the obtuse angle.
In a similar manner, inspecting a less familiar surface, we can
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